The primary results for Michigan and Arizona are in and Mitt Romney won both on Tuesday. The Michigan primary victory, in particular, will help Romney stave off the recent charge by Rick Santorum in the polls.

At press time, no numbers were final, but in Arizona, Mr. Romney was projected to win with about 43 percent of the vote compared with 28 percent for Rick Santorum. The two other contenders, Newt Gingrich and Ron Paul, trailed well behind.

In Michigan, where the contest has been especially heated and polls have swung wildly between Romney and Mr. Santorum in recent weeks, it was a closer battle, with Romney projected to win with about 41 percent of the vote, compared with 38 percent for Santorum. Mr. Paul and Mr. Gingrich lagged behind with about 12 percent and 7 percent of the vote, respectively.

While Arizona’s Romney victory was expected, Michigan – Romney’s home state – was considered a less certain and particularly crucial win for Romney. Winning both states Tuesday helps give him back the aura of inevitability he’s tried to create for himself, which was badly shaken after his losses to Santorum three weeks ago.

“We didn’t win by a lot, but we won by enough, and that’s all that counts,” Romney told a room of cheering supporters in Novi, Mich., Tuesday night. “A week ago, the pundits and pollsters were ready to count us out,” he said. “I was confident that we could come together today and take a giant step toward a brighter future.”

The primaries were the first nominating contests since the ones in Colorado, Minnesota, and Missouri three weeks ago and the last ones before Super Tuesday next week.

While delegates are at stake – 29 delegates in Arizona, where the winner takes all, and 30 delegates in Michigan, which will be awarded proportionately – the primaries have been closely watched mostly as a gauge of how the two front-running candidates are doing.

Losing Michigan is a blow to Santorum, who is already at a disadvantage to Romney in terms of fundraising, staffing, and organization.

This is his first loss in the Midwest and raises questions about whether he can win in a non-caucus state. Failing to deliver a victory in Michigan, despite leading heavily in polls two weeks ago, will also raise renewed questions about his performance as a candidate and his ability to appeal to voters beyond Evangelicals and the far right.

Santorum turned in a shaky debate performance last week and has made several other missteps, including using some odd and religiously toned rhetoric to describe President Obama and suggest the country is degenerating into moral ruin. He said he “almost threw up” in reaction to John F. Kennedy’s 1960 speech on religion, and his robocalls targeting Michigan Democrats, urging them to vote for him in the Republican primary, also sparked controversy.

Some GOP leaders have also questioned his decision to put so much emphasis on social issues, including contraception, which may alienate some women and more-moderate Republicans.

In a speech to supporters Tuesday night, Santorum sought to put a better spin on the evening, calling it an “absolutely great night” due to the close final vote. “A month ago they didn’t know who we are,” he told his supporters. “They do now.” He went on to launch into a long tribute to his mother and wife – emphasizing in particular their professional accomplishments, in a possible response to critics who have suggested he believes women shouldn’t hold career ambitions.

Romney, meanwhile, also faces lingering questions – including why he’s had such a hard time delivering victories in some states, despite his enormous financial and organizational advantages, and how he intends to overcome the lack of enthusiasm he is able to generate among many Republicans.

But Tuesday’s victories – particularly Michigan, which at one point seemed so elusive – will go a long way toward easing some of those doubts. Most importantly, Romney avoided what would have been an embarrassing loss in his home state, and heads into Super Tuesday in a much stronger position.

In his victory speech, Romney largely avoided mentioning Santorum at all and tried to speak as the presumptive nominee, focusing on his economic platform and his differences with Mr. Obama.

“I’m offering a real choice and a very different direction” from Obama, he said.

Despite the wins, exit polls in Michigan also showed some warning signs for Republicans, and a general dissatisfaction with the choices.

Less than half of respondents said that they strongly favor their candidate, while just over half said that they had reservations or disliked the other candidate more.

Not surprisingly, given the strong poll swings in recent weeks, about a quarter of respondents said that they made up their mind in the past few days.